If α and β are the roots of the quadratic equation x 2 + x s i n θ - 2 s i n θ = 0 ,…

If α and β are the roots of the quadratic equation x2+xsinθ-2sinθ=0, θ0,π2 , then α12+β12(α-12+β-12).(α-β)24 is equal to :
  1. 26(sinθ+8)12
  2. 212(sinθ-4)12
  3. 212(sinθ+8)12
  4. 212(sinθ-8)6

Solution

Let E=α12+β12α-12+β-12α-β24=α12+β12α12+β12α12×β12α-β24
E=αβ12α-β24=αβ12α+β2-4αβ12
Now α+β=-sinθ
αβ=-2sinθ
E=-2sinθ12-sinθ2-4-2sinθ12
=2128+sinθ12

Asked in: JEE Main 2019 (10 Apr Shift 1)

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